0
Research Papers

A Machine Learning Approach to Kinematic Synthesis of Defect-Free Planar Four-Bar Linkages

[+] Author and Article Information
Shrinath Deshpande

Computer-Aided Design and Innovation Lab,
Department of Mechanical Engineering,
Stony Brook University,
Stony Brook, NY 11794-2300

Anurag Purwar

Computer-Aided Design and Innovation Lab,
Department of Mechanical Engineering,
Stony Brook University,
Stony Brook, NY 11794-2300
e-mail: anurag.purwar@stonybrook.edu

1Corresponding author.

Manuscript received March 29, 2018; final manuscript received December 12, 2018; published online February 4, 2019. Assoc. Editor: Conrad Tucker.

J. Comput. Inf. Sci. Eng 19(2), 021004 (Feb 04, 2019) (10 pages) Paper No: JCISE-18-1078; doi: 10.1115/1.4042325 History: Received March 29, 2018; Revised December 12, 2018

Synthesizing circuit-, branch-, or order-defects-free planar four-bar mechanism for the motion generation problem has proven to be a difficult problem. These defects render synthesized mechanisms useless to machine designers. Such defects arise from the artificial constraints of formulating the problem as a discrete precision position problem and limitations of the methods, which ignore the continuity information in the input. In this paper, we bring together diverse fields of pattern recognition, machine learning, artificial neural network, and computational kinematics to present a novel approach that solves this problem both efficiently and effectively. At the heart of this approach lies an objective function, which compares the motion as a whole thereby capturing designer's intent. In contrast to widely used structural error or loop-closure equation-based error functions, which convolute the optimization by considering shape, size, position, and orientation of the given task simultaneously, this objective function computes motion difference in a form, which is invariant to similarity transformations. We employ auto-encoder neural networks to create a compact and clustered database of invariant motions of known defect-free linkages, which serve as a good initial choice for further optimization. In spite of highly nonlinear parameters space, our approach discovers a wide pool of defect-free solutions very quickly. We show that by employing proven machine learning techniques, this work could have far-reaching consequences to creating a multitude of useful and creative conceptual design solutions for mechanism synthesis problems, which go beyond planar four-bar linkages.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.

References

McCarthy, J. M. , and Soh, G. S. , 2010, Geometric Design of Linkages, Vol. 11, Springer, New York.
Sandor, G. N. , and Erdman, A. G. , 1997, Advanced Mechanism Design: Analysis and Synthesis, Vol. 2, Prentice Hall, Englewood Cliffs, NJ.
Hunt, K. , 1978, Kinematic Geometry of Mechanisms, Clarendon Press, Oxford.
Hartenberg, R. S. , and Denavit, J. , 1964, Kinematic Synthesis of Linkages, McGraw-Hill, New York.
Suh, C. H. , and Radcliffe, C. W. , 1978, Kinematics and Mechanism Design, Wiley, New York.
Lohse, P. , 2013, Getriebesynthese: Bewegungsablufe Ebener Koppelmechanismen, Springer, Berlin.
Chase, T. , and Mirth, J. , 1993, “ Circuits and Branches of Single-Degree-of-Freedom Planar Linkages,” ASME J. Mech. Des., 115(2), pp. 223–230.
Burmester, L. , 1886, Lehrbuch Der Kinematik, Verlag Von Arthur Felix, Leipzig, Germany.
Cabrera, J. A. , Simon, A. , and Prado, M. , 2002, “ Optimal Synthesis of Mechanisms With Genetic Algorithms,” Mech. Mach. Theory, 37(10), pp. 1165–1177.
Sardashti, A. , Daniali, H. M. , and Varedi, S. M. , 2013, “ Optimal Free-Defect Synthesis of Four-Bar Linkage With Joint Clearance Using PSO Algorithm,” Meccanica, 48(7), pp. 1681–1693.
Ebrahimi, S. , and Payvandy, P. , 2015, “ Efficient Constrained Synthesis of Path Generating Four-Bar Mechanisms Based on the Heuristic Optimization Algorithms,” Mech. Mach. Theory, 85, pp. 189–204.
Bulatovic, R. R. , Dordevic, S. R. , and Dordevic, V. S. , 2013, “ Cuckoo Search Algorithm: A Metaheuristic Approach to Solving the Problem of Optimum Synthesis of a Six-Bar Double Dwell Linkage,” Mech. Mach. Theory, 61, pp. 1–13.
Ullah, I. , and Kota, S. , 1997, “ Optimal Synthesis of Mechanisms for Path Generation Using Fourier Descriptors and Global Search Methods,” ASME J. Mech. Des., 119(4), pp. 504–510.
Wu, J. , Ge, Q. J. , Gao, F. , and Guo, W. Z. , 2011, “ On the Extension of a Fourier Descriptor Based Method for Planar Four-Bar Linkage Synthesis for Generation of Open and Closed Paths,” ASME J. Mech. Rob., 3(3), p. 031002.
Li, X. , Wu, J. , and Ge, Q. J. , 2016, “ A Fourier Descriptor-Based Approach to Design Space Decomposition for Planar Motion Approximation,” ASME J. Mech. Rob., 8(6), p. 064501.
Buskiewicz, J. , Starosta, R. , and Walczak, T. , 2009, “ On the Application of the Curve Curvature in Path Synthesis,” Mech. Mach. Theory, 44(6), pp. 1223–1239.
Khan, N. , Ullah, I. , and Al-Grafi, M. , 2015, “ Dimensional Synthesis of Mechanical Linkages Using Artificial Neural Networks and Fourier Descriptors,” Mech. Sci., 6(1), pp. 29–34.
Mcgarva, J. R. , 1994, “ Rapid Search and Selection of Path Generating Mechanisms From a Library,” Mech. Mach. Theory, 29(2), pp. 223–235.
Wandling, G. R., Sr. , 2000, “ Synthesis of Mechanisms for Function, Path, and Motion Generation Using Invariant Characterization, Storage and Search Methods,” Ph.D. thesis, Iowa State University, Ames, IA. https://lib.dr.iastate.edu/rtd/12383/
Yue, C. , Su, H.-J. , and Ge, Q. , 2011, “ Path Generator Via the Type-P Fourier Descriptor for Open Curves,” 13th World Congress in Mechanism and Machine Science, Guanajuato, Mexico, June 19–25, Paper No. AIL506.
Chu, J. K. , and Sun, J. W. , 2010, “ A New Approach to Dimension Synthesis of Spatial Four-Bar Linkage Through Numerical Atlas Method,” ASME J. Mech. Rob., 2(4), p. 041004.
Cui, M. , Femiani, J. , Hu, J. , Wonka, P. , and Razdan, A. , 2009, “ Curve Matching for Open 2D Curves,” Pattern Recognit. Lett., 30(1), pp. 1–10.
McCarthy, J. M. , 1990, Introduction to Theoretical Kinematics, The MIT Press, Cambridge, MA.
Bottema, O. , and Roth, B. , 1979, Theoretical Kinematics, Dover Publication, New York.
Lewis, J. P. , 1995, “ Fast Normalized Cross-Correlation,” Vision Interface, Vol. 10, Canadian Image Processing and Pattern Recognition Society, pp. 120–123.
Marimont, R. B. , and Shapiro, M. B. , 1979, “ Nearest Neighbor Searches and the Curse of Dimensionality,” J. Inst. Math. Appl., 24(1), pp. 59–70.
Hinton, G. E. , and Salakhutdinov, R. R. , 2006, “ Reducing the Dimensionality of Data With Neural Networks,” Science, 313(5786), pp. 504–507. [PubMed]
Song, C. , Liu, F. , Huang, Y. , Wang, L. , and Tan, T. , 2013, “ Auto-Encoder Based Data Clustering,” Iberoamerican Congress on Pattern Recognition, Springer, Berlin, pp. 117–124.
Ward, J. H. , 1963, “ Hierarchical Grouping to Optimize an Objective Function,” J. Am. Stat. Assoc., 58(301), pp. 236–244.
Ge, Q. J. , Purwar, A. , Zhao, P. , and Deshpande, S. , 2016, “ A Task Driven Approach to Unified Synthesis of Planar Four-Bar Linkages Using Algebraic Fitting of a Pencil of G-Manifolds,” ASME J. Comput. Inf. Sci. Eng., 17(3), p. 031011.
Deshpande, S. , and Purwar, A. , 2017, “ A Task-Driven Approach to Optimal Synthesis of Planar Four-Bar Linkages for Extended Burmester Problem,” ASME J. Mech. Rob., 9(6), p. 061005.

Figures

Grahic Jump Location
Fig. 3

(a) The path of the input motion along with direction of parametrization and (b) motion components x(t), y(t), θ(t) are plotted against parameter t

Grahic Jump Location
Fig. 4

Curvature and its unsigned integral for the path shown in Fig. 3

Grahic Jump Location
Fig. 5

Path and motion signatures of the motion shown in Fig.3: (a) path signature and (b) motion signature

Grahic Jump Location
Fig. 2

The machine learning approach begins by creating an invariant signature for the path and the motion data, which facilitates a compact and hierarchical clustered database and an auto-encoder neural network trained to elicit good, defect-free solutions or subjected to local, fast optimization. The results are defect-free conceptual design solutions for input problems.

Grahic Jump Location
Fig. 1

The four-bar mechanism obtained using the precision position approach suffers from circuit defect, as no coupler circuit passes through all precision positions

Grahic Jump Location
Fig. 6

Part path is formed by trimming whole path followed by translation and scaling. Arrows indicate the increasing direction of parameter t.

Grahic Jump Location
Fig. 7

Path signatures of part p and whole W from Fig. 6. The array index i corresponds to the index location for the array of the K. The domain of path signature is scale-invariant but the range still has a scaling factor, which is taken care of by normalized cross-correlation.

Grahic Jump Location
Fig. 8

Normalized cross correlation of the signatures computed along each direction is shown. It can be seen that exact match is found at j =0.

Grahic Jump Location
Fig. 9

Motion signatures of the trajectories shown in Fig. 6. The domain as well as range of motion signature is invariant to similarity transformation.

Grahic Jump Location
Fig. 10

Dissimilarity function of two motion signatures along both directions. It can be seen that the exact match is found at j =0, where the template is fully embedded inside the other motion.

Grahic Jump Location
Fig. 11

Parametric representation of four-bar linkage with all revolute joints. We set l0 = 1 and one fixed joint at the origin of the global frame along with making the fixed length of four-bar parallel to the x-axis.

Grahic Jump Location
Fig. 12

Coupler motions of the four-bar linkage with variation of parameters l1 and l2. It can be seen that motion topology changes from close-loop Grashof to open-loop Triple Rocker.

Grahic Jump Location
Fig. 13

Motion signatures obtained by steps given in Sec. 2. Although topology difference is even more evident in this representation; it also signifies the similarity pattern between them.

Grahic Jump Location
Fig. 14

Distance (Emin) from Eq. (4) as the parameters l1 and l3 are varied. Although open loop breaks at l1:0.55, l3:1.5, there are no spikes of error function in the region near singularity, as the shape is very similar between the two topologies.

Grahic Jump Location
Fig. 15

Probability distribution function used in random sampling for parameters l1, l2, and l3

Grahic Jump Location
Fig. 16

A small-scale version of the auto-encoder. This network takes five-dimensional input in the input layer. At each encoder layer, the input is compressed into a vector of lower dimensions, the lowest at the bottleneck layer.

Grahic Jump Location
Fig. 20

Case study 2—Query result: motion signatures in the dataset with highest similarity

Grahic Jump Location
Fig. 21

Case study 2: first eight linkages in Table 4 and their resultant coupler motions

Grahic Jump Location
Fig. 17

Case study 1: path traced by hip joint during sit-to-stand motion

Grahic Jump Location
Fig. 18

First eight linkages in Table 2 and their resultant coupler paths

Grahic Jump Location
Fig. 19

Case study 2: user-specified motion necessary for the snow shoveling task

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In